What is the probability of getting a license plate that has a repeated letter or digit if you live in a state that has one numeral followed by two letters followed by four numerals? (Round to the nearest whole percent.)
To determine the probability of getting a license plate with a repeated letter or digit, we need to calculate the number of favorable outcomes (license plates with repeated letters or digits) and divide it by the total number of possible outcomes (all possible license plates).
The total number of possible license plates can be calculated by multiplying the number of options for each position. In this case, we have:
- The first position has 10 options (0-9).
- The second and third positions have 26 options each (A-Z).
- The fourth to seventh positions have 10 options each (0-9).
Therefore, the total number of possible license plates is 10 * 26 * 26 * 10 * 10 * 10 * 10 = 676,000,000.
To calculate the number of license plates with a repeated letter or digit, we need to consider the different cases:
1. Repeated letter in the first and second positions:
- The first position has 26 options (A-Z).
- The second position has 1 option (it must be the same letter as the first position).
- The third position has 25 options (all letters except the one used in the first two positions).
- The fourth to seventh positions each have 10 options (0-9).
Hence, the number of license plates with a repeated letter in the first and second positions is 26 * 1 * 25 * 10 * 10 * 10 * 10 = 6,250,000.
2. Repeated letter in the second and third positions:
- The first position has 10 options (0-9).
- The second position has 26 options (A-Z).
- The third position has 1 option (it must be the same letter as the second position).
- The fourth to seventh positions each have 10 options (0-9).
Therefore, the number of license plates with a repeated letter in the second and third positions is 10 * 26 * 1 * 10 * 10 * 10 * 10 = 2,600,000.
3. Repeated digit in the fourth and fifth positions:
- The first position has 10 options (0-9).
- The second and third positions have 26 options each (A-Z).
- The fourth position has 10 options (0-9).
- The fifth position has 1 option (it must be the same digit as the fourth position).
- The sixth and seventh positions each have 10 options (0-9).
Hence, the number of license plates with a repeated digit in the fourth and fifth positions is 10 * 26 * 26 * 10 * 1 * 10 * 10 = 676,000.
4. Repeated digit in the fifth and sixth positions:
- The first position has 10 options (0-9).
- The second to fourth positions have 26 options each (A-Z).
- The fifth position has 10 options (0-9).
- The sixth position has 1 option (it must be the same digit as the fifth position).
- The seventh position has 10 options (0-9).
Therefore, the number of license plates with a repeated digit in the fifth and sixth positions is 10 * 26 * 26 * 10 * 1 * 10 * 10 = 676,000.
Adding up the cases, we have a total of 6,250,000 + 2,600,000 + 676,000 + 676,000 = 10,202,000 license plates with a repeated letter or digit.
Finally, to calculate the probability, we divide the number of favorable outcomes (license plates with repeated letters or digits) by the total number of outcomes:
P(repeated letter or digit) = (10,202,000 / 676,000,000) * 100 ≈ 1.51%
Therefore, the probability of getting a license plate with a repeated letter or digit is approximately 1.51% (rounded to the nearest whole percent).